cotx+tanx=secx+cscx
To solve the equation cot(x) + tan(x) = sec(x) + csc(x), we can start by manipulating the expression to find a common denominator.
Recall the trigonometric identities:
- cot(x) = 1/tan(x)
- sec(x) = 1/cos(x)
- csc(x) = 1/sin(x)
Using these identities, we can rewrite the equation as:
1/tan(x) + tan(x) = 1/cos(x) + 1/sin(x)
Next, let's find a common denominator for the fractions on the left side of the equation. The common denominator in this case is sin(x) * cos(x). Multiplying each term by this common denominator, we have:
(sin(x) + cos^2(x)) / (sin(x) * cos(x)) = (cos(x) + sin(x)) / (sin(x) * cos(x))
Simplifying the left side and right side of the equation separately:
(sin(x) + cos^2(x)) / (sin(x) * cos(x)) = (cos(x) + sin(x)) / (sin(x) * cos(x))
((sin(x) + cos^2(x)) / (sin(x) * cos(x))) - ((cos(x) + sin(x)) / (sin(x) * cos(x))) = 0
Now, combine the terms on the left side of the equation:
((sin(x) + cos^2(x)) - (cos(x) + sin(x))) / (sin(x) * cos(x)) = 0
sin(x) + cos^2(x) - cos(x) - sin(x) = 0
Simplifying further, we have:
cos^2(x) - cos(x) = 0
Factoring out cos(x):
cos(x) * (cos(x) - 1) = 0
Setting each factor equal to zero:
cos(x) = 0 or cos(x) - 1 = 0
Solving for x:
1) For cos(x) = 0:
x = π/2 + nπ, where n is an integer.
2) For cos(x) - 1 = 0:
cos(x) = 1
x = 2nπ, where n is an integer.
Therefore, the solution to the equation cot(x) + tan(x) = sec(x) + csc(x) is:
x = π/2 + nπ, x = 2nπ for integer values of n.
To solve the equation cot(x) + tan(x) = sec(x) + csc(x), we can start by simplifying both sides of the equation using trigonometric identities.
First, let's recall some trigonometric identities:
- cot(x) = 1/tan(x)
- sec(x) = 1/cos(x)
- csc(x) = 1/sin(x)
Now, let's substitute these identities in the equation:
1/tan(x) + tan(x) = 1/cos(x) + 1/sin(x)
To combine the terms with a common denominator, multiply the first term by sin(x)/sin(x) and the second term by cos(x)/cos(x):
sin(x)/[sin(x) * cos(x)] + (tan^2(x) * cos(x))/[sin(x) * cos(x)] = cos(x)/[cos(x) * sin(x)] + (cos(x) * sin(x))/[cos(x) * sin(x)]
Simplifying further:
[sin(x) + tan^2(x) * cos(x)]/[sin(x) * cos(x)] = [cos(x) + cos(x) * sin(x)]/[cos(x) * sin(x)]
Now, we can cross-multiply:
(sin(x) + tan^2(x) * cos(x)) * (cos(x) * sin(x)) = (cos(x) + cos(x) * sin(x)) * (sin(x) * cos(x))
Expanding both sides:
(sin^2(x) * cos(x) * sin(x) + tan^2(x) * cos^2(x) * sin(x)) = (sin(x) * cos(x) + cos^2(x) * sin^2(x))
We can simplify this equation further by using additional trigonometric identities:
- sin^2(x) + cos^2(x) = 1
- tan^2(x) = sec^2(x) - 1
Applying these identities:
(sin(x) * cos(x) * sin(x) + (sec^2(x) - 1) * cos^2(x) * sin(x)) = (sin(x) * cos(x) + cos^2(x) * sin^2(x))
Simplifying:
sin^2(x) * cos(x) * sin(x) + sec^2(x) * cos^2(x) * sin(x) - cos^2(x) * sin(x) = sin(x) * cos(x) + cos^2(x) * sin^2(x)
Now, let's gather like terms on one side of the equation:
sin^2(x) * cos(x) * sin(x) + sec^2(x) * cos^2(x) * sin(x) - cos^2(x) * sin(x) - sin(x) * cos(x) - cos^2(x) * sin^2(x) = 0
Next, let's simplify further by factoring out sin(x):
sin(x) * [sin^2(x) * cos(x) + sec^2(x) * cos^2(x) - cos^2(x) - cos(x) - cos^2(x) * sin(x)] = 0
Now, we have a product of two factors equal to zero. To find the solutions, we can set each factor equal to zero:
1) sin(x) = 0:
This gives us x = 0, π, 2π, ...
2) sin^2(x) * cos(x) + sec^2(x) * cos^2(x) - cos^2(x) - cos(x) - cos^2(x) * sin(x) = 0:
We can rewrite this equation as:
cos(x)(sin^2(x) + sec^2(x) * cos(x) - cos(x) - (1 + sin^2(x)))
Simplifying further:
cos(x)(sin^2(x) + cos^2(x) * sec(x) - cos(x) - 1)
Using the identity sec^2(x) = 1 + tan^2(x):
cos(x)(sin^2(x) + cos^2(x) * (1 + tan^2(x)) - cos(x) - 1)
Expanding and simplifying:
cos(x)(sin^2(x) + cos^2(x) + cos^2(x) * tan^2(x) - cos(x) - 1)
cos(x)(1 + 1 - cos(x) - 1 + cos^2(x) * tan^2(x))
cos(x)(2 - cos(x) + cos^2(x) * tan^2(x))
We can now solve this equation using the solutions for cos(x) = 0 and 2 - cos(x) + cos^2(x) * tan^2(x) = 0. After obtaining the values of x from these solutions, we can check them in the original equation cot(x) + tan(x) = sec(x) + csc(x) to verify if they satisfy the equation.